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The given problem involves graphing the phase diagram for the system of homogeneous linear differential equations defined by:

\[
\begin{pmatrix}
x' \\
y'
\end{pmatrix}
=
\begin{pmatrix}
-1 & -2 \\
1 & -4
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
\]

with initial conditions:

\[
\begin{pmatrix}
x(0) \\
y(0)
\end{pmatrix}
=
\begin{pmatrix}
1 \\
4
\end{pmatrix}
\]

### To solve the system:
1. **Eigenvalues and Eigenvectors**: First, compute the eigenvalues and eigenvectors of the coefficient matrix:

\[
A = \begin{pmatrix}
-1 & -2 \\
1 & -4
\end{pmatrix}
\]

2. **Solve the differential system**: The solution to the system can be written as a combination of exponential terms based on the eigenvalues and eigenvectors.
   
3. **Phase Portrait**: Graph the solutions in the phase plane by plotting trajectories based on the general solution.

I'll begin by finding the eigenvalues and eigenvectors of the matrix and proceed with the solution. Let's start the calculations.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you need assistance with a manual method for solving this system, feel free to ask, and I can walk you through the process.

Here are a few related questions to expand on this topic:
1. What are the steps for solving a 2x2 system of differential equations?
2. How do you interpret the phase plane for a linear system of differential equations?
3. What role do eigenvalues play in determining the stability of a system?
4. How can you use initial conditions to determine the specific solution to a system?
5. What are the possible types of phase portraits for a system of linear differential equations?

**Tip:** When solving systems of differential equations, always look for symmetry in the matrix, as this can often simplify calculations of eigenvalues and eigenvectors.

Graph the phase diagram for the system of homogeneous linear differential equations defined by: x' = [-1 -2; 1 -4] * [x; y] with initial conditions: x(0) = 1, y(0) = 4.

Learn how to graph the phase diagram of a system of homogeneous linear differential equations using matrix techniques, eigenvalues, and initial conditions. This problem covers topics in linear algebra and differential equations, suitable for university-level mathematics.

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